Lanczos-based methods for matrix functions

Abstract

We study Lanczos-based methods for tasks involving matrix functions. We begin by resurfacing a range of ideas regarding matrix-free quadrature which, to the best of our knowledge, have not been treated simultaneously. This enables the development of a unified perspective from which a number of commonly used randomized methods for spectrum and spectral sum approximation can be understood. We proceed to develop optimal Krylov subspace methods for approximating the product of a rational matrix function with a fixed vector. Finally, we show how the optimality of such methods can be used to obtain fine-grained spectrum dependent bounds for standard Lanczos-based methods for approximating a wide class of matrix functions applied to a vector.